The remainders when three positive integers are divided by 5 are 1, 2, and 3.  Find the remainder when their product is divided by 5.
The product of the three integers is equivalent in modulo 5 to the product of the modulo 5 residues of the three integers.  We multiply these residues to find the remainder: $$ 1 \cdot 2 \cdot 3 \equiv 6 \equiv \boxed{1} \pmod{5}. $$